We analyse in a systematic way the (non-)compact four dimensional Einstein-Weyl spaces equipped with a Bianchi metric. We show that Einstein-Weyl structures with a Class A Bianchi metric have a conformal scalar curvature of constant sign on the manifold. Moreover, we prove that most of them are conformally Einstein or conformally Kähler ; in the non-exact Einstein-Weyl case with a Bianchi metri...

We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational eld with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new eld equation generalizing the notion of Killing spinors. The solutions of this spinor eld equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Di...

The object of the present paper is to characterize Cotton tensor on a 3-dimensional Sasakian manifold admitting $\eta$-Ricci solitons. After introduction, we study manifolds and introduce new notion, namely, pseudo-symmetric manifolds. Next deal with 3-manifold Among others prove that such constant scalar curvature Einstein some appropriate conditions. Also, classify nature soliton metric. Fina...

The object of the present paper is to study some properties Kenmotsu manifold whose metric conformal $\eta$-Einstein soliton. We have studied certain admitting also constructed a 3-dimensional satisfying

The curvature tensor \(\hat{R}\) of a manifold is called harmonic, if it obeys the condition \(\Delta^{\text{(HR)}}\hat{R}=0\), where \(\Delta^{\text{(HR)}}=DD^{\ast} +
 D^{\ast}D\) Hodge–deRham Laplacian. It proved that all solutions Einstein equations in vacuum, as well Einstein–Cartan theory vacuum have harmonic curvature. statement only Einstein’s type \(N\) (describing gravitational r...

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V , as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M . In addition we show that the differential or variation dV of V , or equivalently the variation of the L norm of the Weyl curvature, is intrinsically determined by the conformal inf...

Abstract We study conformal $$\eta$$ ? -Einstein solitons on the framework of trans-Sasakian manifold in dimension three. Existence is discussed. Then we find some results which are where Ricci tensor cyclic parallel and Codazzi type. also consider curvature conditions with addition to manifold. use torse-for...

Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold M,g,f,μ to be Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that λ,n+m-Einstein M,g,w having harmonic Weyl tensor, ∇jw∇mwCjklm=0 and ∇lw∇lw<0 reduces fibers. Finally, if φ=−m∇lnw φRic-vector field on M an ψ=∇w ψRic-vector...

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete manifold with fourth-order divergence-free Weyl tensor zero radial curvature is locally warped product $(n-1)$-dimensional Einstein fibers, provided the potential function proper. As consequence, result about nonexistence of mult...

A generalized theory unifying gravity with electromagnetism was proposed by Einstein in 1945. He considered a Hermitian metric on a real space-time. In this work we review Einstein’s idea and generalize it further to consider gravity in a complex Hermitian space-time. email: [email protected] Published in ”Einstein in Alexandria, The Scientific Symposium”, Editor Edward Witten, Publisher Bibilot...